English
An alternative formulation of the product expansion for c.minpoly under a normal extension.
Русский
Альтернативная формулировка разложения минимального полинома через произведение над носителем.
LaTeX
$$$ c.minpoly.map(\\operatorname{algebraMap} K L) = \\prod_{x \\in c.carrier.Elem} (X - C x) $$$
Lean4
/-- If an element `x` is a root of a nonzero polynomial `p`, then the degree of `p` is at least the
degree of the minimal polynomial of `x`. See also `minpoly.IsIntegrallyClosed.degree_le_of_ne_zero`
which relaxes the assumptions on `A` in exchange for stronger assumptions on `B`. -/
theorem degree_le_of_ne_zero {p : A[X]} (pnz : p ≠ 0) (hp : Polynomial.aeval x p = 0) :
degree (minpoly A x) ≤ degree p :=
calc
degree (minpoly A x) ≤ degree (p * C (leadingCoeff p)⁻¹) := min A x (monic_mul_leadingCoeff_inv pnz) (by simp [hp])
_ = degree p := degree_mul_leadingCoeff_inv p pnz
